Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$. It follows from the spectral radius formula that if $M\in\mathcal B_n$ is not nilpotent, then $\rho(M)$, the spectral radius of $M$, is $\ge1$.

I need a condition for $\rho(M)>1$ in terms of the proportion of 1s.

QUESTION. Is there a $c\in(0,1)$ such that for any $n\ge1$ and any $M\in\mathcal B_n$ with the number of 0s less than $cn^2$ we have $\rho(M)>1$?

If so, what's the best known value of $c$?